Word can do this if you hit Paste, then click on the weird little symbol you see at the end of the text you paste and choose "Match Destination Formatting" or "Keep Text Only". (The little symbol--it looks a bit like a crude drawing of a clipboard--is actually a popup menu. It disappears as soon as you start to type again.)
Just before I saw your comment, I had been trolling through Word Help on just the subject of special pasting and ran across discussion regarding the operation of the Paste Options button (which always popped up under that operation but which I hadn't paid any attention to since it disappeared as soon as any other operation was initiated).
That little button does exactly what I want (ie, matching cut text to destination formatting, namely my default preferences) and it does it slicker than .... I was going to report on same here, when I came across your suggestion.
That pretty much seals it. Word now wins hands down (at least for cutting and pasting HTML and other differently formatted texts).